Displacement operator

The displacement operator for one mode in quantum optics is the shift operator

${\hat {D}}(\alpha )=\exp \left(\alpha {\hat {a}}^{\dagger }-\alpha ^{\ast }{\hat {a}}\right)$,

where $\alpha$ is the amount of displacement in optical phase space, $\alpha ^{*}$ is the complex conjugate of that displacement, and ${\hat {a}}$ and ${\hat {a}}^{\dagger }$ are the lowering and raising operators, respectively. The name of this operator is derived from its ability to displace a localized state in phase space by a magnitude $\alpha$. It may also act on the vacuum state by displacing it into a coherent state. Specifically, ${\hat {D}}(\alpha )|0\rangle =|\alpha \rangle$ where $|\alpha \rangle$ is a coherent state, which is the eigenstates of the annihilation (lowering) operator.

Properties

The displacement operator is a unitary operator, and therefore obeys ${\hat {D}}(\alpha ){\hat {D}}^{\dagger }(\alpha )={\hat {D}}^{\dagger }(\alpha ){\hat {D}}(\alpha )={\hat {1}}$, where ${\hat {1}}$ is the identity operator. Since ${\hat {D}}^{\dagger }(\alpha )={\hat {D}}(-\alpha )$, the hermitian conjugate of the displacement operator can also be interpreted as a displacement of opposite magnitude ($-\alpha$). The effect of applying this operator in a similarity transformation of the ladder operators results in their displacement.

${\hat {D}}^{\dagger }(\alpha ){\hat {a}}{\hat {D}}(\alpha )={\hat {a}}+\alpha$

${\hat {D}}(\alpha ){\hat {a}}{\hat {D}}^{\dagger }(\alpha )={\hat {a}}-\alpha$

The product of two displacement operators is another displacement operator, apart from a phase factor, has the total displacement as the sum of the two individual displacements. This can be seen by utilizing the Baker-Campbell-Hausdorff formula.

$e^{{\alpha {\hat {a}}^{{\dagger }}-\alpha ^{*}{\hat {a}}}}e^{{\beta {\hat {a}}^{{\dagger }}-\beta ^{*}{\hat {a}}}}=e^{{(\alpha +\beta ){\hat {a}}^{{\dagger }}-(\beta ^{*}+\alpha ^{*}){\hat {a}}}}e^{{(\alpha \beta ^{*}-\alpha ^{*}\beta )/2}}.$

which shows us that:

${\hat {D}}(\alpha ){\hat {D}}(\beta )=e^{{(\alpha \beta ^{*}-\alpha ^{*}\beta )/2}}{\hat {D}}(\alpha +\beta )$

When acting on an eigenket, the phase factor $e^{{(\alpha \beta ^{*}-\alpha ^{*}\beta )/2}}$ appears in each term of the resulting state, which makes it physically irrelevant.^[1]

Alternative expressions

Two alternative ways to express the displacement operator are:

${\hat {D}}(\alpha )=e^{{-{\frac {1}{2}}|\alpha |^{2}}}e^{{+\alpha {\hat {a}}^{{\dagger }}}}e^{{-\alpha ^{{*}}{\hat {a}}}}$

${\hat {D}}(\alpha )=e^{{+{\frac {1}{2}}|\alpha |^{2}}}e^{{-\alpha ^{{*}}{\hat {a}}}}e^{{+\alpha {\hat {a}}^{{\dagger }}}}$

Multimode displacement

The displacement operator can also be generalized to multimode displacement. A multimode creation operator can be defined as

${\hat A}_{{\psi }}^{{\dagger }}=\int d{\mathbf {k}}\psi ({\mathbf {k}}){\hat a}^{{\dagger }}({\mathbf {k}})$,

where $\mathbf {k}$ is the wave vector and its magnitude is related to the frequency $\omega _{{{\mathbf {k}}}}$ according to $|{\mathbf {k}}|=\omega _{{{\mathbf {k}}}}/c$. Using this definition, we can write the multimode displacement operator as

${\hat {D}}_{{\psi }}(\alpha )=\exp \left(\alpha {\hat A}_{{\psi }}^{{\dagger }}-\alpha ^{\ast }{\hat A}_{{\psi }}\right)$,

and define the multimode coherent state as

$|\alpha _{{\psi }}\rangle \equiv {\hat {D}}_{{\psi }}(\alpha )|0\rangle$.

References

1. Christopher Gerry and Peter Knight: Introductory Quantum Optics. Cambridge (England): Cambridge UP, 2005.

See also

• Optical Phase Space